Spectra of subdivision operators

Let a := {a(k)}(k is an element of Z) be a sequence of complex numbers and a(k) = 0 except for finitely many k. The subdivision operator Sa associated with a is the bi-infinite matrix S-a := (a(j - 2k))(j,k is an element of Z ). This operator plays an important role in wavelet analysis and subdivisio n algorithms. As the adjoint it is closely related to the well-known transf er operators (also called Ruelle operator). In this paper we show that for any 1 less than or equal to p less than or e qual to infinity, the spectrum of S-a in l(p)(Z) is always a closed disc ce ntered at the origin. Moreover, except for finitely many points, all the po ints in the open disc of the spectrum lie in the residual spectrum.